Face covers and rooted minors in bounded genus graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912270924120064 |
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| author | Fiorini, Samuel Kober, Stefan Seweryn, Michał T. Shantanam, Abhinav Yuditsky, Yelena |
| author_facet | Fiorini, Samuel Kober, Stefan Seweryn, Michał T. Shantanam, Abhinav Yuditsky, Yelena |
| contents | A {\em rooted graph} is a graph together with a designated vertex subset, called the {\em roots}. In this paper, we consider rooted graphs embedded in a fixed surface. A collection of faces of the embedding is a {\em face cover} if every root is incident to some face in the collection. We prove that every $3$-connected, rooted graph that has no rooted $K_{2,t}$ minor and is embedded in a surface of Euler genus $g$, has a face cover whose size is upper-bounded by some function of $g$ and $t$, provided that the face-width of the embedding is large enough in terms of $g$. In the planar case, we prove an unconditional $O(t^4)$ upper bound, improving a result of Böhme and Mohar~\cite{BM02}. The higher genus case was claimed without a proof by Böhme, Kawarabayashi, Maharry and Mohar~\cite{BKMM08}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_09230 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Face covers and rooted minors in bounded genus graphs Fiorini, Samuel Kober, Stefan Seweryn, Michał T. Shantanam, Abhinav Yuditsky, Yelena Combinatorics Discrete Mathematics 05C10, 05C83 A {\em rooted graph} is a graph together with a designated vertex subset, called the {\em roots}. In this paper, we consider rooted graphs embedded in a fixed surface. A collection of faces of the embedding is a {\em face cover} if every root is incident to some face in the collection. We prove that every $3$-connected, rooted graph that has no rooted $K_{2,t}$ minor and is embedded in a surface of Euler genus $g$, has a face cover whose size is upper-bounded by some function of $g$ and $t$, provided that the face-width of the embedding is large enough in terms of $g$. In the planar case, we prove an unconditional $O(t^4)$ upper bound, improving a result of Böhme and Mohar~\cite{BM02}. The higher genus case was claimed without a proof by Böhme, Kawarabayashi, Maharry and Mohar~\cite{BKMM08}. |
| title | Face covers and rooted minors in bounded genus graphs |
| topic | Combinatorics Discrete Mathematics 05C10, 05C83 |
| url | https://arxiv.org/abs/2503.09230 |